Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

72 Differentiation of Functions [C/t. 2
2. Principal properties
of differentials.
1) dc = 0, where c = const.
2) d*- Ax, where x is an independent variable.
3) d(cu) = cdu.
4) d(u v) = du dv.
5) d (uv) udv + v du.
7)
3. Applying the differential to approximate calculations. If the increment
A* of the argument x is small in absolute value, then the differential dy of the
function y = f(x) and the increment At/ of the function are approximately
equal:
that is,
whence
A# =^ dy,
Example 3. By how much (approximately) does the side of a square change
if its area increases from 9 m 2 to 9.1 m 2 ?
Solution. If x is the area of the square and y is its side, then
It is given that # = 9 and A*
The increment At/ in the side
0.1.
of the square may be calculated approximately
as follows:
ky^zdy--=y' Ax = j=z -0.1 = 0. 016m.
4. Higher-order differentials. A second-order differential
of a first-order differential:
We similarly define the differentials of the third and higher orders.
If y = f(x) and x is an independent variable, then
But if y = /(), where w = cp(x), then
d*y = y"' (du) 9 + 3y" du d*u + y' d'u
and so forth. (Here the primes denote derivatives with respect to M).
is the differential
712. Find the increment Ay and the differentia! dy of the function
# = 5* -f x 2
for x = 2 and A# = 0.001.